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<span class="md-ellipsis">
Before starting
Before Starting
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<span class="md-nav__icon md-icon"></span>
Before starting
Before Starting
</label>
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<span class="md-ellipsis">
0.1 About this book
0.1 About This Book
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<span class="md-ellipsis">
0.2 How to read
0.2 How to Use This Book
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<span class="md-ellipsis">
Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
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Chapter 1. Encounter with algorithms
Chapter 1. Encounter With Algorithms
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<span class="md-ellipsis">
1.1 Algorithms are everywhere
1.1 Algorithms Are Everywhere
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<span class="md-ellipsis">
1.2 What is an algorithm
1.2 What Is an Algorithm
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<span class="md-ellipsis">
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
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<span class="md-nav__icon md-icon"></span>
Chapter 2. Complexity analysis
Chapter 2. Complexity Analysis
</label>
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<span class="md-ellipsis">
2.1 Algorithm efficiency assessment
2.1 Algorithm Efficiency Evaluation
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<span class="md-ellipsis">
2.2 Iteration and recursion
2.2 Iteration and Recursion
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<span class="md-ellipsis">
2.3 Time complexity
2.3 Time Complexity
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<span class="md-ellipsis">
2.4 Space complexity
2.4 Space Complexity
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<span class="md-ellipsis">
Chapter 3. Data structures
Chapter 3. Data Structures
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<span class="md-nav__icon md-icon"></span>
Chapter 3. Data structures
Chapter 3. Data Structures
</label>
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<span class="md-ellipsis">
3.1 Classification of data structures
3.1 Classification of Data Structures
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<span class="md-ellipsis">
3.2 Basic data types
3.2 Basic Data Types
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<span class="md-ellipsis">
3.3 Number encoding *
3.3 Number Encoding *
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<span class="md-ellipsis">
3.4 Character encoding *
3.4 Character Encoding *
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<span class="md-ellipsis">
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
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<span class="md-nav__icon md-icon"></span>
Chapter 4. Array and linked list
Chapter 4. Array and Linked List
</label>
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<span class="md-ellipsis">
4.2 Linked list
4.2 Linked List
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<span class="md-ellipsis">
4.4 Memory and cache *
4.4 Memory and Cache *
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<a href="#1-key-review" class="md-nav__link">
<span class="md-ellipsis">
1. &nbsp; Key review
1. &nbsp; Key Review
</span>
</a>
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<span class="md-ellipsis">
Chapter 5. Stack and queue
Chapter 5. Stack and Queue
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Chapter 5. Stack and queue
Chapter 5. Stack and Queue
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<span class="md-ellipsis">
5.3 Double-ended queue
5.3 Double-Ended Queue
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<span class="md-ellipsis">
Chapter 6. Hash table
Chapter 6. Hashing
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<span class="md-nav__icon md-icon"></span>
Chapter 6. Hash table
Chapter 6. Hashing
</label>
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<span class="md-ellipsis">
6.1 Hash table
6.1 Hash Table
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<span class="md-ellipsis">
6.2 Hash collision
6.2 Hash Collision
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<span class="md-ellipsis">
6.3 Hash algorithm
6.3 Hash Algorithm
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<span class="md-ellipsis">
7.1 Binary tree
7.1 Binary Tree
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<span class="md-ellipsis">
7.2 Binary tree traversal
7.2 Binary Tree Traversal
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<span class="md-ellipsis">
7.3 Array Representation of tree
7.3 Array Representation of Tree
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<span class="md-ellipsis">
7.4 Binary Search tree
7.4 Binary Search Tree
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<span class="md-ellipsis">
7.5 AVL tree *
7.5 AVL Tree *
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<span class="md-ellipsis">
8.2 Building a heap
8.2 Building a Heap
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<span class="md-ellipsis">
8.3 Top-k problem
8.3 Top-K Problem
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<span class="md-ellipsis">
9.2 Basic graph operations
9.2 Basic Operations on Graphs
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<span class="md-ellipsis">
9.3 Graph traversal
9.3 Graph Traversal
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<span class="md-ellipsis">
10.1 Binary search
10.1 Binary Search
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<span class="md-ellipsis">
10.2 Binary search insertion
10.2 Binary Search Insertion
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<span class="md-ellipsis">
10.3 Binary search boundaries
10.3 Binary Search Edge Cases
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<span class="md-ellipsis">
10.4 Hashing optimization strategies
10.4 Hash Optimization Strategy
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<span class="md-ellipsis">
10.5 Search algorithms revisited
10.5 Search Algorithms Revisited
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<span class="md-ellipsis">
11.1 Sorting algorithms
11.1 Sorting Algorithms
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<span class="md-ellipsis">
11.2 Selection sort
11.2 Selection Sort
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<span class="md-ellipsis">
11.3 Bubble sort
11.3 Bubble Sort
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<span class="md-ellipsis">
11.4 Insertion sort
11.4 Insertion Sort
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<span class="md-ellipsis">
11.5 Quick sort
11.5 Quick Sort
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<span class="md-ellipsis">
11.6 Merge sort
11.6 Merge Sort
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<span class="md-ellipsis">
11.7 Heap sort
11.7 Heap Sort
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<span class="md-ellipsis">
11.8 Bucket sort
11.8 Bucket Sort
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<span class="md-ellipsis">
11.9 Counting sort
11.9 Counting Sort
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<span class="md-ellipsis">
11.10 Radix sort
11.10 Radix Sort
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<span class="md-ellipsis">
Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
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Chapter 12. Divide and conquer
Chapter 12. Divide and Conquer
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<span class="md-ellipsis">
12.1 Divide and conquer algorithms
12.1 Divide and Conquer Algorithms
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<span class="md-ellipsis">
12.2 Divide and conquer search strategy
12.2 Divide and Conquer Search Strategy
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<span class="md-ellipsis">
12.3 Building binary tree problem
12.3 Building a Binary Tree Problem
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<span class="md-ellipsis">
12.4 Tower of Hanoi Problem
12.4 Hanoi Tower Problem
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<span class="md-ellipsis">
13.1 Backtracking algorithms
13.1 Backtracking Algorithm
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<span class="md-ellipsis">
13.2 Permutation problem
13.2 Permutations Problem
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<span class="md-ellipsis">
13.3 Subset sum problem
13.3 Subset-Sum Problem
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<span class="md-ellipsis">
13.4 n queens problem
13.4 N-Queens Problem
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<span class="md-ellipsis">
Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
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Chapter 14. Dynamic programming
Chapter 14. Dynamic Programming
</label>
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<span class="md-ellipsis">
14.1 Introduction to dynamic programming
14.1 Introduction to Dynamic Programming
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<span class="md-ellipsis">
14.2 Characteristics of DP problems
14.2 Characteristics of Dynamic Programming Problems
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<span class="md-ellipsis">
14.3 DP problem-solving approach
14.3 Dynamic Programming Problem-Solving Approach
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<span class="md-ellipsis">
14.4 0-1 Knapsack problem
14.4 0-1 Knapsack Problem
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<span class="md-ellipsis">
14.5 Unbounded knapsack problem
14.5 Unbounded Knapsack Problem
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<span class="md-ellipsis">
14.6 Edit distance problem
14.6 Edit Distance Problem
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<span class="md-ellipsis">
15.1 Greedy algorithms
15.1 Greedy Algorithm
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<span class="md-ellipsis">
15.2 Fractional knapsack problem
15.2 Fractional Knapsack Problem
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15.3 Maximum capacity problem
15.3 Maximum Capacity Problem
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<span class="md-ellipsis">
15.4 Maximum product cutting problem
15.4 Maximum Product Cutting Problem
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16.1 Installation
16.1 Programming Environment Installation
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<span class="md-ellipsis">
16.2 Contributing
16.2 Contributing Together
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16.3 Terminology
16.3 Terminology Table
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1. &nbsp; Key review
1. &nbsp; Key Review
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<!-- Page content -->
<h1 id="45-summary">4.5 &nbsp; Summary<a class="headerlink" href="#45-summary" title="Permanent link">&para;</a></h1>
<h3 id="1-key-review">1. &nbsp; Key review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<h3 id="1-key-review">1. &nbsp; Key Review<a class="headerlink" href="#1-key-review" title="Permanent link">&para;</a></h3>
<ul>
<li>Arrays and linked lists are two basic data structures, representing two storage methods in computer memory: contiguous space storage and non-contiguous space storage. Their characteristics complement each other.</li>
<li>Arrays support random access and use less memory; however, they are inefficient in inserting and deleting elements and have a fixed length after initialization.</li>
<li>Linked lists implement efficient node insertion and deletion through changing references (pointers) and can flexibly adjust their length; however, they have lower node access efficiency and consume more memory.</li>
<li>Common types of linked lists include singly linked lists, circular linked lists, and doubly linked lists, each with its own application scenarios.</li>
<li>Lists are ordered collections of elements that support addition, deletion, and modification, typically implemented based on dynamic arrays, retaining the advantages of arrays while allowing flexible length adjustment.</li>
<li>The advent of lists significantly enhanced the practicality of arrays but may lead to some memory space wastage.</li>
<li>During program execution, data is mainly stored in memory. Arrays provide higher memory space efficiency, while linked lists are more flexible in memory usage.</li>
<li>Caches provide fast data access to CPUs through mechanisms like cache lines, prefetching, spatial locality, and temporal locality, significantly enhancing program execution efficiency.</li>
<li>Due to higher cache hit rates, arrays are generally more efficient than linked lists. When choosing a data structure, the appropriate choice should be made based on specific needs and scenarios.</li>
<li>Arrays and linked lists are two fundamental data structures, representing two different ways data can be stored in computer memory: contiguous memory storage and scattered memory storage. The characteristics of the two complement each other.</li>
<li>Arrays support random access and use less memory; however, inserting and deleting elements is inefficient, and the length is immutable after initialization.</li>
<li>Linked lists achieve efficient insertion and deletion of nodes by modifying references (pointers), and can flexibly adjust length; however, node access is inefficient and memory consumption is higher. Common linked list types include singly linked lists, circular linked lists, and doubly linked lists.</li>
<li>A list is an ordered collection of elements that supports insertion, deletion, search, and modification, typically implemented based on dynamic arrays. It retains the advantages of arrays while allowing flexible adjustment of length.</li>
<li>The emergence of lists has greatly improved the practicality of arrays, but may result in some wasted memory space.</li>
<li>During program execution, data is primarily stored in memory. Arrays provide higher memory space efficiency, while linked lists offer greater flexibility in memory usage.</li>
<li>Caches provide fast data access to the CPU through mechanisms such as cache lines, prefetching, and spatial and temporal locality, significantly improving program execution efficiency.</li>
<li>Because arrays have higher cache hit rates, they are generally more efficient than linked lists. When choosing a data structure, appropriate selection should be made based on specific requirements and scenarios.</li>
</ul>
<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>: Does storing arrays on the stack versus the heap affect time and space efficiency?</p>
<p>Arrays stored on both the stack and heap are stored in contiguous memory spaces, and data operation efficiency is essentially the same. However, stacks and heaps have their own characteristics, leading to the following differences.</p>
<p><strong>Q</strong>: Does storing an array on the stack versus on the heap affect time efficiency and space efficiency?</p>
<p>Arrays stored on the stack and on the heap are both stored in contiguous memory space, so data operation efficiency is basically the same. However, the stack and heap have their own characteristics, leading to the following differences.</p>
<ol>
<li>Allocation and release efficiency: The stack is a smaller memory block, allocated automatically by the compiler; the heap memory is relatively larger and can be dynamically allocated in the code, more prone to fragmentation. Therefore, allocation and release operations on the heap are generally slower than on the stack.</li>
<li>Size limitation: Stack memory is relatively small, while the heap size is generally limited by available memory. Therefore, the heap is more suitable for storing large arrays.</li>
<li>Flexibility: The size of arrays on the stack needs to be determined at compile-time, while the size of arrays on the heap can be dynamically determined at runtime.</li>
<li>Allocation and deallocation efficiency: The stack is a relatively small piece of memory, with allocation automatically handled by the compiler; the heap is relatively larger and can be dynamically allocated in code, more prone to fragmentation. Therefore, allocation and deallocation operations on the heap are usually slower than on the stack.</li>
<li>Size limitations: Stack memory is relatively small, and the heap size is generally limited by available memory. Therefore, the heap is more suitable for storing large arrays.</li>
<li>Flexibility: The size of an array on the stack must be determined at compile time, while the size of an array on the heap can be determined dynamically at runtime.</li>
</ol>
<p><strong>Q</strong>: Why do arrays require elements of the same type, while linked lists do not emphasize same-type elements?</p>
<p>Linked lists consist of nodes connected by references (pointers), and each node can store data of different types, such as int, double, string, object, etc.</p>
<p>In contrast, array elements must be of the same type, allowing the calculation of offsets to access the corresponding element positions. For example, an array containing both int and long types, with single elements occupying 4 bytes and 8 bytes respectively, cannot use the following formula to calculate offsets, as the array contains elements of two different lengths.</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Element memory address = array memory address + element length * element index</span>
<p><strong>Q</strong>: Why do arrays require elements of the same type, while linked lists do not emphasize this requirement?</p>
<p>Linked lists are composed of nodes, with nodes connected through references (pointers), and each node can store different types of data, such as <code>int</code>, <code>double</code>, <code>string</code>, <code>object</code>, etc.</p>
<p>In contrast, array elements must be of the same type, so that the corresponding element position can be obtained by calculating the offset. For example, if an array contains both <code>int</code> and <code>long</code> types, with individual elements occupying 4 bytes and 8 bytes respectively, then the following formula cannot be used to calculate the offset, because the array contains two different "element lengths".</p>
<div class="highlight"><pre><span></span><code><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a><span class="c1"># Element Memory Address = Array Memory Address (first Element Memory address) + Element Length * Element Index</span>
</code></pre></div>
<p><strong>Q</strong>: After deleting a node, is it necessary to set <code>P.next</code> to <code>None</code>?</p>
<p>Not modifying <code>P.next</code> is also acceptable. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been effectively removed from the list, and where <code>P</code> points no longer affects the list.</p>
<p>From a garbage collection perspective, for languages with automatic garbage collection mechanisms like Java, Python, and Go, whether node <code>P</code> is collected depends on whether there are still references pointing to it, not on the value of <code>P.next</code>. In languages like C and C++, we need to manually free the node's memory.</p>
<p><strong>Q</strong>: In linked lists, the time complexity for insertion and deletion operations is <code>O(1)</code>. But searching for the element before insertion or deletion takes <code>O(n)</code> time, so why isn't the time complexity <code>O(n)</code>?</p>
<p>If an element is searched first and then deleted, the time complexity is indeed <code>O(n)</code>. However, the <code>O(1)</code> advantage of linked lists in insertion and deletion can be realized in other applications. For example, in the implementation of double-ended queues using linked lists, we maintain pointers always pointing to the head and tail nodes, making each insertion and deletion operation <code>O(1)</code>.</p>
<p><strong>Q</strong>: In the figure "Linked List Definition and Storage Method", do the light blue storage nodes occupy a single memory address, or do they share half with the node value?</p>
<p>The figure is just a qualitative representation; quantitative analysis depends on specific situations.</p>
<p><strong>Q</strong>: After deleting node <code>P</code>, do we need to set <code>P.next</code> to <code>None</code>?</p>
<p>It is not necessary to modify <code>P.next</code>. From the perspective of the linked list, traversing from the head node to the tail node will no longer encounter <code>P</code>. This means that node <code>P</code> has been removed from the linked list, and it doesn't matter where node <code>P</code> points to at this time—it won't affect the linked list.</p>
<p>From a data structures and algorithms perspective (problem-solving), not disconnecting the pointer doesn't matter as long as the program logic is correct. From the perspective of standard libraries, disconnecting is safer and the logic is clearer. If not disconnected, assuming the deleted node is not properly reclaimed, it may affect the memory reclamation of its successor nodes.</p>
<p><strong>Q</strong>: In a linked list, the time complexity of insertion and deletion operations is <span class="arithmatex">\(O(1)\)</span>. However, both insertion and deletion require <span class="arithmatex">\(O(n)\)</span> time to find the element; why isn't the time complexity <span class="arithmatex">\(O(n)\)</span>?</p>
<p>If the element is first found and then deleted, the time complexity is indeed <span class="arithmatex">\(O(n)\)</span>. However, the advantage of <span class="arithmatex">\(O(1)\)</span> insertion and deletion in linked lists can be demonstrated in other applications. For example, a deque is well-suited for linked list implementation, where we maintain pointer variables always pointing to the head and tail nodes, with each insertion and deletion operation being <span class="arithmatex">\(O(1)\)</span>.</p>
<p><strong>Q</strong>: In the diagram "Linked List Definition and Storage Methods", does the light blue pointer node occupy a single memory address, or does it share equally with the node value?</p>
<p>This diagram is a qualitative representation; a quantitative representation requires analysis based on the specific situation.</p>
<ul>
<li>Different types of node values occupy different amounts of space, such as int, long, double, and object instances.</li>
<li>The memory space occupied by pointer variables depends on the operating system and compilation environment used, usually 8 bytes or 4 bytes.</li>
<li>Different types of node values occupy different amounts of space, such as <code>int</code>, <code>long</code>, <code>double</code>, and instance objects, etc.</li>
<li>The amount of memory space occupied by pointer variables depends on the operating system and compilation environment used, usually 8 bytes or 4 bytes.</li>
</ul>
<p><strong>Q</strong>: Is adding elements to the end of a list always <code>O(1)</code>?</p>
<p>If adding an element exceeds the list length, the list needs to be expanded first. The system will request a new memory block and move all elements of the original list over, in which case the time complexity becomes <code>O(n)</code>.</p>
<p><strong>Q</strong>: The statement "The emergence of lists greatly improves the practicality of arrays, but may lead to some memory space wastage" - does this refer to the memory occupied by additional variables like capacity, length, and expansion multiplier?</p>
<p>The space wastage here mainly refers to two aspects: on the one hand, lists are set with an initial length, which we may not always need; on the other hand, to prevent frequent expansion, expansion usually multiplies by a coefficient, such as <span class="arithmatex">\(\times 1.5\)</span>. This results in many empty slots, which we typically cannot fully fill.</p>
<p><strong>Q</strong>: In Python, after initializing <code>n = [1, 2, 3]</code>, the addresses of these 3 elements are contiguous, but initializing <code>m = [2, 1, 3]</code> shows that each element's <code>id</code> is not consecutive but identical to those in <code>n</code>. If the addresses of these elements are not contiguous, is <code>m</code> still an array?</p>
<p>If we replace list elements with linked list nodes <code>n = [n1, n2, n3, n4, n5]</code>, these 5 node objects are also typically dispersed throughout memory. However, given a list index, we can still access the node's memory address in <code>O(1)</code> time, thereby accessing the corresponding node. This is because the array stores references to the nodes, not the nodes themselves.</p>
<p>Unlike many languages, in Python, numbers are also wrapped as objects, and lists store references to these numbers, not the numbers themselves. Therefore, we find that the same number in two arrays has the same <code>id</code>, and these numbers' memory addresses need not be contiguous.</p>
<p><strong>Q</strong>: The <code>std::list</code> in C++ STL has already implemented a doubly linked list, but it seems that some algorithm books don't directly use it. Is there any limitation?</p>
<p>On the one hand, we often prefer to use arrays to implement algorithms, only using linked lists when necessary, mainly for two reasons.</p>
<p><strong>Q</strong>: Is appending an element at the end of a list always <span class="arithmatex">\(O(1)\)</span>?</p>
<p>If appending an element exceeds the list length, the list must first be expanded before adding. The system allocates a new block of memory and moves all elements from the original list to it, in which case the time complexity becomes <span class="arithmatex">\(O(n)\)</span>.</p>
<p><strong>Q</strong>: "The emergence of lists has greatly improved the practicality of arrays, but may result in some wasted memory space"—does this space waste refer to the memory occupied by additional variables such as capacity, length, and expansion factor?</p>
<p>This space waste mainly has two aspects: on one hand, lists typically set an initial length, which we may not need to fully utilize; on the other hand, to prevent frequent expansion, expansion generally multiplies by a coefficient, such as <span class="arithmatex">\(\times 1.5\)</span>. As a result, there will be many empty positions that we typically cannot completely fill.</p>
<p><strong>Q</strong>: In Python, after initializing <code>n = [1, 2, 3]</code>, the addresses of these 3 elements are contiguous, but initializing <code>m = [2, 1, 3]</code> reveals that each element's id is not continuous; rather, they are the same as those in <code>n</code>. Since the addresses of these elements are not contiguous, is <code>m</code> still an array?</p>
<p>If we replace list elements with linked list nodes <code>n = [n1, n2, n3, n4, n5]</code>, usually these 5 node objects are also scattered throughout memory. However, given a list index, we can still obtain the node memory address in <span class="arithmatex">\(O(1)\)</span> time, thereby accessing the corresponding node. This is because the array stores references to nodes, not the nodes themselves.</p>
<p>Unlike many languages, numbers in Python are wrapped as objects, and lists store not the numbers themselves, but references to the numbers. Therefore, we find that the same numbers in two arrays have the same id, and the memory addresses of these numbers need not be contiguous.</p>
<p><strong>Q</strong>: C++ STL has <code>std::list</code> which has already implemented a doubly linked list, but it seems that some algorithm books don't use it directly. Is there a limitation?</p>
<p>On one hand, we often prefer to use arrays for implementing algorithms and only use linked lists when necessary, mainly for two reasons.</p>
<ul>
<li>Space overhead: Since each element requires two additional pointers (one for the previous element and one for the next), <code>std::list</code> usually occupies more space than <code>std::vector</code>.</li>
<li>Cache unfriendly: As the data is not stored continuously, <code>std::list</code> has a lower cache utilization rate. Generally, <code>std::vector</code> performs better.</li>
<li>Space overhead: Since each element requires two additional pointers (one for the previous element and one for the next element), <code>std::list</code> typically consumes more space than <code>std::vector</code>.</li>
<li>Cache unfriendliness: Since data is not stored contiguously, <code>std::list</code> has lower cache utilization. In general, <code>std::vector</code> has better performance.</li>
</ul>
<p>On the other hand, linked lists are primarily necessary for binary trees and graphs. Stacks and queues are often implemented using the programming language's <code>stack</code> and <code>queue</code> classes, rather than linked lists.</p>
<p><strong>Q</strong>: Does initializing a list <code>res = [0] * self.size()</code> result in each element of <code>res</code> referencing the same address?</p>
<p>No. However, this issue arises with two-dimensional arrays, for example, initializing a two-dimensional list <code>res = [[0]] * self.size()</code> would reference the same list <code>[0]</code> multiple times.</p>
<p><strong>Q</strong>: In deleting a node, is it necessary to break the reference to its successor node?</p>
<p>From the perspective of data structures and algorithms (problem-solving), it's okay not to break the link, as long as the program's logic is correct. From the perspective of standard libraries, breaking the link is safer and more logically clear. If the link is not broken, and the deleted node is not properly recycled, it could affect the recycling of the successor node's memory.</p>
<p>On the other hand, cases where linked lists are necessary mainly involve binary trees and graphs. Stacks and queues usually use the <code>stack</code> and <code>queue</code> provided by the programming language, rather than linked lists.</p>
<p><strong>Q</strong>: Does the operation <code>res = [[0]] * n</code> create a 2D list where each <code>[0]</code> is independent?</p>
<p>No, they are not independent. In this 2D list, all the <code>[0]</code> are actually references to the same object. If we modify one element, we will find that all corresponding elements change accordingly.</p>
<p>If we want each <code>[0]</code> in the 2D list to be independent, we can use <code>res = [[0] for _ in range(n)]</code> to achieve this. The principle of this approach is to initialize <span class="arithmatex">\(n\)</span> independent <code>[0]</code> list objects.</p>
<p><strong>Q</strong>: Does the operation <code>res = [0] * n</code> create a list where each integer 0 is independent?</p>
<p>In this list, all integer 0s are references to the same object. This is because Python uses a caching mechanism for small integers (typically -5 to 256) to maximize object reuse and improve performance.</p>
<p>Although they point to the same object, we can still independently modify each element in the list. This is because Python integers are "immutable objects". When we modify an element, we are actually switching to a reference of another object, rather than changing the original object itself.</p>
<p>However, when list elements are "mutable objects" (such as lists, dictionaries, or class instances), modifying an element directly changes the object itself, and all elements referencing that object will have the same change.</p>
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